VIBRATION OF THE LONDONM
MILLENNIUM FOOTBRIDGE:
PART 2 - CURE

David E Newland

Abstract

Part 2 of this paper considers how much damping is needed to ensure that pedestrian footbridges do not
experience excessive lateral vibration and describes how the necessary damping was provided for the London
Millennium footbridge.

INTRODUCTION

People-excited lateral bridge vibration is likely to occur for pedestrian bridges which have low natural
frequencies of swaying movement (less than 3 Hz) and for which the lateral modes have light damping. In the
case of London’s Millennium Bridge, both these conditions applied.

One solution would have been to stiffen the bridge to increase its natural frequencies and take these
outside the excitation frequency range. However the artistic design of the bridge would have been compromised
by stiffening and this was regarded as most undesirable. The alternative was to find a way of increasing the
bridge’s inherently low damping so that self-excitation did not occur. It has been found that, below a
threshold damping level, motion would build up, but that above the threshold damping level, self-excitation
would not occur. Determining what this threshold level was and then providing a means of introducing the
required amount of added damping proved a challenging task. It has involved adding 37 linear viscous dampers
and over 50 tuned mass vibration absorbers to the initial structure. As a result, this bridge is now probably
the most complex passively-damped structure in the world.

Figure 3 Centre span of the Millennium Bridge

Installing dampers in a way that was consistent with the aesthetic design of the bridge was difficult. A
great deal of effort had been put into choosing a preferred design and the concept of “a blade of
light” had been adopted and received widespread approval (see Deyan Sudjic (ed), Blade of Light:
The Story of London’s Millennium Bridge, 2001). So far as possible dampers had to be mounted
underneath the bridge deck so that they would be out-of-sight of everyone using the bridge.

RECONCILIATION WITH ARUP’S DAMPING CALCULATION

At page 20 of Fitzpatrick et al (2001), Arup give their formula corresponding to (16) as

(17)

Although this has been arrived at by a completely different approach, it is identical with (16). This can
be verified by making the substitutions , , , , and . Apart from different normalisation of the modal shape function, the
main difference is that Arup defined feedback force as proportional to velocity whereas the above analysis
begins by assuming that the feedback force is proportional to displacement (at a fixed frequency). Arup use
their symbol k not for stiffness but to relate pedestrian feedback force to deck lateral velocity,
whereas as
defined above relates feedback force to deck lateral displacement.

Arup’s computation of their proportionality factor k was done by measuring the acceleration
time history under conditions of steady-state crowd loading with a constant number of people walking steadily
over each span (in turn) at the correct speed to resonate with the relevant mode. From this time-history,
they calculated modal velocity (see Fitzpatrick et al, 2001, p. 14). If F is the amplitude of the
modal feedback force (which is assumed proportional to velocity), D is the amplitude of the modal
damping force (also proportional to velocity and known from previous measurements), and V is the
amplitude of the modal velocity, then for conservation of energy,

(18)

so that

(19)

where m is modal mass, and A is the amplitude of the modal acceleration, since
. By plotting
calculated
from (19) (N is the number of people on the span) against modal velocity V, Arup arrived at an
average value for k. Some typical values are shown in figure 4 in which physical rather than modal
results are plotted. The approximately linear relationship in figure 4 appears to derive from the combined
action of two factors: the force per person increases (slowly) with amplitude and more people synchronise
with deck movement at larger amplitudes (see the additional material in Fitzpatrick et al, 2001). Linearity
is of course a starting assumption for the feedback model in part 1.

DEPENDENCE OF NET DAMPING ON THE NUMBER OF PEDESTRIANS

It follows from (16) and (17) that the net loss factor will decrease in proportion to the number of walkers
on the span. If is the number of pedestrians on the span when the damping decreases
to zero so that , the net loss factor for this mode when there are pedestrians on the span will
be

(18)

PRACTICAL DAMPING MEASURES

Based on these considerations, Arup decided to aim for 15% to 20% of critical damping for all lateral and
lateral/torsional modes below 1.5 Hz and 5% to 10% of critical damping for vertical and vertical/torsional
modes below 3 Hz. This is a huge increase in the original damping ratios of these modes which were typically
1% or less. To understand how this was achieved, it is necessary to understand the construction of the
bridge. This can be seen from the photograph, figure 3.

The bridge deck is carried on lateral supports spaced periodically. These reach out to clamp onto the four
parallel steel cables at each side of the deck. To a first approximation, the bridge vibrates like a taut
string passing over supports at the two bridge piers and anchored to fixed supports at the river banks.
Therefore lateral vibration involves shearing of the deck structure with no appreciable bending. All the low
frequency modes have nodes at the attachment of the cables to the bridge piers and to the river bank
anchorages. Although linear viscous dampers can be connected between the bridge deck and these fixed
anchorages, the relative motion here is small and dampers fixed here cannot be made to work efficiently.
Maximum relative shear displacement occurs between the lateral supports near anti-nodes, and therefore away
from the fixed anchorages.

The solution adopted was to fit A-shaped frames to alternate lateral supports, with the points of two
A’s meeting at the intermediate supports (as shown in figure 5).

Between the points of each pair of A frames, a linear viscous damper was mounted. It was possible to do
this so that the moving parts were supported vertically on the upper-side of the lateral supports. All the
viscous dampers were supplied by the US firm Taylor Devices, Inc. and incorporated metal bellows seals so
that they are fully-sealed to the environment.

For the centre span, the damping introduced by frame-mounted viscous dampers was supplemented by the
action of 4 pairs of laterally-acting tuned-mass vibration absorbers supplied by the German firm Gerb
Schwingungsisolierungen GmbH, mounted

on the upper side of the bridge deck’s lateral supports, as shown in figure 5.

An additional 26 pairs of vertically-acting tuned-mass vibration absorbers were installed in similar
positions on other lateral supports to increase vertical damping. This is to guard against the (unlikely)
possibility that synchronous vertical vibration might occur when the lateral problem had been removed. The
tuned-mass vibration absorbers have masses between 1 and 3 tonnes and they are located as close as possible
to the antinodes of the modes that they are damping.

EXPERIMENTAL VALIDATION

In addition to a range of laboratory tests to study human gait and the interaction of pedestrians and moving
platforms, the main experimental tests were carried out on the bridge. These consisted of two essentially
different types of tests. Tests with no people, using a mechanical shaker to provide excitation, were carried
out to measure modal frequencies and damping. This was done initially for the bare bridge, and then for the
bridge with specimen viscous and tuned-mass dampers installed, to verify their action. Tests with walking
people consisted mostly of recirculating tests where a metered number of pedestrians walked in one direction
across a single span, and then immediately turned round and walked back to their starting point. Results from
these tests were used to generate data like that in figure 4 and to confirm the essentially unstable feature
of lateral synchronous excitation. A typical result for the north span, without any added damping, is shown
in figure 6. A metered number of people were instructed to walk steadily at the speed needed to synchronise
with the first lateral mode of the north span. Progressively the number of people walking was increased as
shown by the staircase graph. The bridge deck acceleration (plotted below the staircase graph) increased
slightly until 166 people were walking, when there was a sudden increase in deck lateral response which was
sufficiently violent to stop the test. Since, when fully-laden, the north span can accommodate perhaps 700
people, the reason for the problems on opening day is apparent.

The performance targets for the modified bridge were expressed as rms acceleration levels measured at the
quarter and half-span points with a 1 minute averaging time. The lateral target, after filtering with a
passband of 0.2 to 2.4 Hz was that the rms should not exceed 25x10-3g laterally; the vertical
target in a passband of 0.2 to 4.8 Hz was that the rms should not exceed 50x10-3g vertically.
These targets were to be met in the presence of a test in which 2,000 people walked over the bridge three
times at 0.6 m/s, 0.9 m/s and 1.2 m/s approximately with the bridge comfortably full of people. A great deal
of planning went into the organisation and carrying out of this test which was successfully completed on 30
January 2002. Measured acceleration levels were substantially below the target limits for all the tests,
typically less than one sixth of the agreed limits.

Figure 6. Onset of instability in crowd test on undamped north span,
fundamental mode. As the number of people walking on the span (upper graph) increased to 166 progressively,
the bridge lateral acceleration (lower graph)increased only slowly until instability was reached. The scale
for acceleration is not given, but the peak acceleration reached was about 80x10-3 g at the
right-hand side (Arup figure from Deyan Sudjic (ed), 2001, p. 93)

CONCLUSIONS

The introduction of damping by a combination of frame-mounted viscous dampers and tuned-mass vibration
absorbers has cured the London Millennium Bridge’s famous wobble. It was caused by synchronous lateral
excitation from pedestrians, a phenomenon that was not well-known at the time but for which there is now a
good understanding and good data.

ACKNOWLEDGEMENTS

The bridge’s engineering designers were the Ove Arup Partnership and they designed, tested and
supervised the construction of the vibration control system described in this paper. The author’s role
was as independent technical advisor to the London Millennium Bridge Trust (the bridge’s principal
funding body) for the duration of the remedial work described above. During this period, July 2000 to January
2002, he was pleased to work with the Ove Arup Partnership, and with the W. S. Atkins Group who were advising
the London Borough of Southwark and the Corporation of London. Two detailed papers by Ove Arup about the
bridge are listed below. The interpretation of experimental results in terms of the feedback model described
in part 1 of this paper is original to the author.